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Nominal Logic: A First Order Theory of Names and Binding
 Information and Computation
, 2001
"... This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal L ..."
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Cited by 218 (15 self)
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This paper formalises within firstorder logic some common practices in computer science to do with representing and reasoning about syntactical structures involving named bound variables (as opposed to nameless terms, explicit substitutions, or higher order abstract syntax). It introduces Nominal Logic, a version of firstorder manysorted logic with equality containing primitives for renaming via nameswapping and for freshness of names, from which a notion of binding can be derived. Its axioms express...
Alphastructural recursion and induction
 Journal of the ACM
, 2006
"... The nominal approach to abstract syntax deals with the issues of bound names and αequivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically i ..."
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Cited by 56 (6 self)
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The nominal approach to abstract syntax deals with the issues of bound names and αequivalence by considering constructions and properties that are invariant with respect to permuting names. The use of permutations gives rise to an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax modulo αequivalence. At the heart of this approach is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higherorder logic of principles of αstructural recursion and induction for αequivalence classes from the ordinary versions of these principles for abstract syntax trees.
A Definitional Approach to Primitive Recursion over Higher Order Abstract Syntax
 In Proceedings of the 2003 workshop on Mechanized
, 2003
"... Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K. ..."
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Cited by 27 (5 self)
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Syntax S. J. Ambler (S.Ambler@mcs.le.ac.uk) R. L. Crole (R.Crole@mcs.le.ac.uk) & A. Momigliano (A.Momigliano@mcs.le.ac.uk) Department of Mathematics and Computer Science, University of Leicester, Leicester, LE1 7RH, U.K.
A Dependent Type Theory with Names and Binding
 In Proceedings of the 2004 Computer Science Logic Conference, number 3210 in Lecture notes in Computer Science
, 2004
"... We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on FraenkelMostowski (FM) set theory how to address this through firstclass names: in this paper we present a dependent type theory for prog ..."
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Cited by 18 (1 self)
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We consider the problem of providing formal support for working with abstract syntax involving variable binders. Gabbay and Pitts have shown in their work on FraenkelMostowski (FM) set theory how to address this through firstclass names: in this paper we present a dependent type theory for programming and reasoning with such names. Our development is based on a categorical axiomatisation of names, with freshness as its central notion. An associated adjunction captures constructions known from FM theory: the freshness quantifier N , namebinding, and unique choice of fresh names. The Schanuel topos  the category underlying FM set theory  is an instance of this axiomatisation.
Permissive nominal terms and their unification: an infinite, coinfinite approach to nominal techniques
, 2010
"... ..."
Alphastructural recursion and induction (Extended Abstract)
 THEOREM PROVING IN HIGHER ORDER LOGICS, 18TH INTERNATIONAL CONFERENCE, TPHOLS 2005, OXFORD UK, AUGUST 2005, PROCEEDINGS, VOLUME 3603 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... There is growing evidence for the usefulness of name permutations when dealing with syntax involving names and namebinding. In particular they facilitate an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax tr ..."
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Cited by 6 (2 self)
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There is growing evidence for the usefulness of name permutations when dealing with syntax involving names and namebinding. In particular they facilitate an attractively simple formalisation of common, but often technically incorrect uses of structural recursion and induction for abstract syntax trees modulo αequivalence. At the heart of this formalisation is the notion of finitely supported mathematical objects. This paper explains the idea in as concrete a way as possible and gives a new derivation within higherorder logic of principles of αstructural recursion and induction for αequivalence classes from the ordinary versions of these principles for abstract syntax trees.
Finite and infinite support in nominal algebra and logic: nominal completeness theorems for free
 Journal of Symbolic Logic
, 2012
"... By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissivenominal models, so the construction hinges on generating from an insta ..."
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Cited by 2 (1 self)
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By operations on models we show how to relate completeness with respect to permissivenominal models to completeness with respect to nominal models with finite support. Models with finite support are a special case of permissivenominal models, so the construction hinges on generating from an instance of the latter, some instance of the former in which sufficiently many inequalities are preserved between elements. We do this using an infinite generalisation of nominal atomsabstraction. The results are of interest in their own right, but also, we factor the mathematics so as to maximise the chances that it could be used offtheshelf for other nominal reasoning systems too. Models with infinite support can be easier to work with, so it is useful to have a semiautomatic theorem to transfer results from classes of infinitelysupported nominal models to the more restricted class of models with finite support. In conclusion, we consider different permissivenominal syntaxes and nominal models and discuss how they relate to the results proved here.
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"... Permissive nominal terms and their unification: an infinite, coinfinite approach to nominal techniques ..."
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Permissive nominal terms and their unification: an infinite, coinfinite approach to nominal techniques
unknown title
, 2010
"... Twolevel nominal sets and semantic nominal terms: an extension of nominal set theory for handling metavariables ..."
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Twolevel nominal sets and semantic nominal terms: an extension of nominal set theory for handling metavariables